What Is My Auto Insurer Required To Do In This Scenario?

Location: Houston, TX

I was rear-ended at a dead stop in Houston. Other party determined to be 100% at fault. The at-fault driver and I have the same auto insurer. I was injured and am currently completing treatment. My vehicle, a 2012 something, was in perfectly fine mechanical condition prior to the accident. It was drivable from the scene so I drove it home. My air conditioner stopped working--it was blowing but suddenly not cold anymore, and this was noticeable even on the way home. Again, my car was in A-OK mechanical condition. My air was working just fine and had been for years without issue. I have an attorney for the personal injury side and that's squared away so far. But apparently I'm on my own with the property damage claim.

The other guy's side (my same insurer) is paying for the repairs to the back of the vehicle; however, they are fighting me tooth and nail about my air conditioner. I have zero intention of accepting back my vehicle without air conditioning. Their client broke my air conditioner, therefore the insurer is liable for it. They're trying every dumb deflection they can think of, like full on "someone who has never seen your car has determined we are not paying since the collision was in the rear and the air conditioner is located in the front". I'm not a physics expert, but I'm pretty sure kinetic energy can move from one place to another if you smack into something really hard with a large vehicle at a decent speed?

Anyway, they are continuing to refuse to pay or even look into it. The shop volunteered to at least check the relay and fuse--both fine. They are checking the coolant tomorrow morning and said they would see if they can visually see a leak or something but they're not going further than that, I guess? The insurer, after I mercilessly mocked the stupid things they said, now say that if a mechanic makes a report that says it was the cause, then they will pay.

My problem is, the mechanic is suggesting I could get stuck with the bill for any major diagnostics if the findings aren't what I expect. But it seems to me that it's the insurer's job to look into the damage I report and find out what's wrong with it at their expense, no? I feel like it shouldn't be up to me to beg some mechanic to find a leak or whatever the issue is and be willing to put it in writing. It's almost like they are coming at me like some scammer who wants her air conditioner fixed for free on someone else's dime, and it's up to me to prove I'm not lying and that I wasn't in fact driving around in Houston this summer wearing all black office clothes without a working air conditioner. No air conditioner is not an option here, it is heat-stroke hot and I commute two hours every day and driving without air conditioning would be miserable and dangerous. Why should I tolerate this extreme discomfort through no fault of my own?

Anyway, my question is: Is the insurer obligated to look into the broken air conditioner, how far must they go to find the issue, and can they stick me with the bill if a mechanic won't say what they need to hear? Why is my air conditioner being treated differently than the body damage?